Perturbation theory and calculus of variations (840G1)

15 credits, Level 6

Spring teaching

The aim of this module is to introduce you to a variety of techniques, involving primarily ordinary differential equations, that have applications in several branches of mathematics applied to the natural and social sciences. Once a practical problem is modelled and the relevant ordinary differential equations are constructed, techniques such as dimensional analysis and scaling are used to check that the model is correct and to reduce the equations to their simplest form.

Since many of these equations do not have explicit/closed-form solutions, the module will introduce regular and singular perturbation methods which give rise to approximate solutions.

The final part of the module introduces concepts from the calculus of variations which studies functions whose input is another function rather than a numerical value, known as functionals. The goal is to then find the extremals of these functionals, that is functions that minimise or maximise the output of the functional, with many real-world problems falling under this setting. For example, the isoperimetric problem which aims to determine a plane figure of the largest possible area whose boundary has a specified length is a special application of the calculus of variations.


100%: Lecture


20%: Coursework (Portfolio, Problem set)
80%: Examination (Unseen examination)

Contact hours and workload

This module is approximately 150 hours of work. This breaks down into about 33 hours of contact time and about 117 hours of independent study. The University may make minor variations to the contact hours for operational reasons, including timetabling requirements.

We regularly review our modules to incorporate student feedback, staff expertise, as well as the latest research and teaching methodology. We’re planning to run these modules in the academic year 2023/24. However, there may be changes to these modules in response to feedback, staff availability, student demand or updates to our curriculum. We’ll make sure to let you know of any material changes to modules at the earliest opportunity.


This module is offered on the following courses: